• Alexander Aleksandrov: Residue theory and logarithmic de Rham complex
  The purpose of the talk is to give an elementary introduction to the theory of residue of logarithmic differential forms, and to describe some of the less known applications of this theory, developed by the author in the past few years. In particular, we briefly discuss the notion of residue due to H. Poincare, J. de Rham, J. Leray and K. Saito, and then obtain a nice description of regular meromorphic differential forms in terms of residues of logarithmic differential forms. We also discuss a new method for computing the topological index of complex vector fields on hypersurfaces with arbitrary singularities, some applications to the theory of holonomic D-modules of Fuchsian and logarithmic types, to the theory of Hodge structures on singular varieties, etc.
• Paolo Aluffi: Chern classes of hyperplane arrangements
  We relate invariants such as the Chern-Schwartz-MacPherson class of the class in the Grothendieck group of varieties to combinatorial invariants, for the case of hyperplane arrangements in projective space. We will explain the connection between these algebro-geometric invariants and results of Orlik-Solomon, Mustata-Schenck, and Wakefield-Yoshinaga.
• Donu Arapura: Hodge theory and the fundamental group
  I want to discuss some thoughts about Hodge and motivic structures associated to the fundamental group of an algebraic variety using Tannakian methods. I will compare this to other approaches due to Hain, Morgan, Simpson and others.
• Markus Banagl: Singularities and Intersection Spaces: Theory and Application.
  In many situations, it is homotopy theoretically possible to associate to a singular space in a natural way a generalized geometric Poincare complex, called its intersection space, whose cohomology turns out to be a new cohomology theory for singular spaces, not isomorphic in general to intersection cohomology or Cheeger-L²-cohomology. An alternative description of the new theory by a de Rham complex of global differential forms is available. The talk will discuss the properties of the new theory, stressing differences to intersection cohomology, as well as indicate applications of these methods, even outside singularity theory. We will consider the K-theory of intersection spaces. We will see how, as a by-product, one obtains results on equivariant cohomology and flat bundles. The theory also addresses questions in type II string theory and mirror symmetry. While intersection cohomology is stable under small resolutions, the new theory is often stable under deformations of singularities. The latter result is partly joint work with L. Maxim. An analytic description remains to be found, but we shall indicate a partial result based on results of Melrose and Hausel-Hunsicker-Mazzeo.
• Jean-Paul Brasselet: Some insights on the Euler local obstruction
  The lecture concerns a joint work with N. Grulha and M. Ruas. The local Euler obstruction was introduced by R. MacPherson as a key ingredient for his construction of characteristic classes of singular complex algebraic varieties. Then, an equivalent definition was given by J.-P. Brasselet and M.-H. Schwartz using vector fields and many other definitions and interpretations have been provided. The notion of local Euler obstruction developed mainly in two directions: the first one comes back to MacPherson's definition and concerns with differential forms. That is developed by W. Ebeling and S. Gusein-Zade in a series of papers. The second one relates local Euler obstruction with functions and maps defined on the variety. That approach is useful to relate local Euler obstruction with other indices. Aim of the lecture is to present these new features on the subject.
• Sylvain Cappell: Symmetries and rigidity of aspherical manifolds
  Classical work of Borel had shown that an action of the circle on a manifold with contractible universal cover yields non-trivial center in the manifold's fundamental group. In the 60's, Conner and Raymond made further deep investigations which led them to conjecture a converse to Borel's result. We construct counter-examples to this conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater than or equal to 6) which have non-trivial center in their fundamental groups but no circle actions. The constructions and invariants involve synthesizing disparate ideas of geometric topology of singular spaces, geometric group theory and hyperbolic geometry. (This is joint work with Shmuel Weinberger and Min Yan.)
• Zhi Chen: KZ connections and Brauer type algebras
  We introduce a Brauer type algebra and a BMW type algebra for every Coxeter group and every pseudo reflection group. This is an application of KZ connections on the complementary spaces of hyperplane arrangements.
• Manuel Gonzalez Villa : Motivic zeta functions for quasi-ordinary hypersurface singularities
  Denef and Loeser applied the theory of motivic integration to define new invariants of hypersurface singularities. The motivic Milnor fibre, introduced as a formal limit of the motivic zeta function, encodes much of the geometry of the Milnor fibration. We discuss using toric methods how to compute these invariants for the class of irreducible hypersurface quasi-ordinary singularities. We prove that these invariants are determined by the embedded topological type of the singularity, which is known to be completely characterized by a finite number of characteristic monomials of some fractional power series associated to the hypersurface This is a joint work with P.D. Gonzalez-Perez.
• Jose Ignacio Cogolludo: Mordell-Weil groups of elliptic threefolds over P², Alexander polynomials, and quasi-toric relations of curves
  Given C a (possibly reducible and non-reduced) projective plane curve, we define the Alexander polynomial A(t) of C with respect to the multiplicities of its components. To each root of A(t) one associates a cyclic covering of P² ramified along C and an elliptic threefold W with a natural automorphism. The purpose of this talk will be to establish a connection between the multiplicity of roots of A(t), the rank of the Mordell-Weil group of W, and the existence of enough "independent" maps onto elliptic orbifold curves (a generalization of pencils) containing C as fibers. Curves for which such pencils exist are a generalization of torus-type curves and are called here quasi-toric curves. This connection allows one to use the theory of elliptic curves over a function field to find properties of quasi-projective groups and Alexander polynomials. In particular one can find bounds for the degree of Alexander polynomials of curves with only nodes and cusps as singularities. This is a joint work with Anatoly Libgober.
• Guangfeng Jiang: The Global Invariant of Hyperplane Arrangements
  Abstract.
• Ludmil Katzarkov: Degenerations, singularities and wall crossings
  In this talk we will introduce new categorical structures from classical prospective of degnerations.
• Toshitake Kohno: Monodromy groups of conformal field theory
  There is an action of the mapping class groups on the space of the conformal blocks for Riemann surfaces defined by monodromy. We give a qualitative estimate for the images of such representations of mapping class groups. In particular, we show that the image of any Johnson subgroup contains a non-abelian free group. In the case of braid groups we describe the monodromy group in relation with triangle groups. Based on the estimate of the monodromy groups we give an answer to conjectures by Squier on Burau representations of braid groups. It was shown by P. Gilmer and G. Masbaum that the monodromy groups of conformal field theory for Riemann surfaces are defined over cyclotomic integers. We show that, in general, the monodromy groups are not isomorphic to a higher rank irreducible lattices in semi-simple Lie groups. This is a joint work with Louis Funar.
• Zhi Lu: A differential operator and tom Dieck-Kosniowski-Stong localization theorem
  In this talk, we define a differential operator on the "dual" algebra of the unoriented G-representation algebra introduced by Conner and Floyd, where G is a mod 2 torus group of rank n. With the help of G-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simply equivalent description of tom Dieck--Kosniowski--Stong localization theorem in the setting of smooth closed n-manifolds with effective smooth G-actions (also called n-dimensional 2-torus manifolds). Some applications will be considered. This is a joint work with Qiangbo Tan.
• Teresa Monteiro Fernandes: Propagation for solutions with moderate growth of D-Modules
  We obtain a Cauchy Theorem for holomorphic solutions of D-modules with moderate growth conditions in a space of parameters and obtain an estimate for the obstruction to the propagation (microsupport).
• Mutsuo Oka: Mixed polynomials and mixed varieties
  Mixed varieties is a complex analytic technique to study real algebraic variety of codimension 2. I will explain basic properties of mixed varieties and some applications.
• Piotr Pragacz: On positivity of Thom polynomials
  The pioneering papers of Griffiths and Fulton and Lazarsfeld investigated numerical positivity related to ample vector bundles in differential and algebraic geometry. Their various variants are nowadays widely investigated in algebraic geometry. Among main objects of global singularity theory are the Thom polynomials of singularity classes. We shall consider Thom polynomials of singularities of mappings and Lagrangian and Legendrian Thom polynomials. We shall show that in some bases coming from representation theory, they admit positive expansions. (This is a report on joint work with M. Mikosz and A. Weber.)
• Richard Rimanyi: Thom series via equivariant localization and iterated residues
  Thom polynomials measure how topology forces singularities. Natural infinite sequences of Thom polynomials can be arranged in formal power series, the Thom series. In the talk we will explore different interpretations and computational strategies of Thom series, as well as present some open problems. This is a joint work with L. Feher, and we will also report on recent results of Berczi-Szenes and Kazarian.
• Christian Schnell: Hodge modules on abelian varieties
  I will explain some results on Hodge modules on abelian varieties, and applications to the study of irregular varieties. (Joint work with Mihnea Popa.)
• Jörg Schürmann: Characteristic classes of Hilbert schemes of points via symmetric products.
  We explain a new formula for the generating series of the Hirzebruch characteristic homology classes of the Hilbert schemes of points for a smooth quasi-projective variety, push-forward to the corresponding symmetric products. This result is based on two facts: (i) a nice interplay between the geometric definition of a motivic power structure and a motivic Pontrjagin ring of the symmetric products, (ii) a formula for the generating series of the Hirzebruch characteristic homology classes of the symmetric products. This is joint work in progress with L. Maxim, S. Cappell, T. Ohmoto and S. Yokura.
• Kiyoshi Takeuchi: Motivic Milnor fibers and Newton polyhedra
  By computing the equivariant mixed Hodge numbers of motivic Milnor fibers introduced by Denef-Loeser etc., we obtain various formulae for the Jordan normal forms of the local and global monodromies of polynomials. Especially we focus our attention on the global ones, i.e., the monodromies at infinity. For polynomials over affine complete intersection varieties the results will be described by the mixed volumes of the faces of their Newton polyhedra. This is a joint work with Y. Matsui and A. Esterov.
• Andrzej Weber: Local equivariant Chern classes of Singular varieties
  Equivariant cohomology is a powerful tool to study of complex manifolds equipped with a torus action. The localization theorem of Atiyah and Bott and the resulting formula of Berline-Vergne allow to compute global invariants of singular subsets in terms of the fixed points of the action. We will concentrate on the Chern(-Schwartz-MacPherson) classes. The global class is determined by the local contributions coming from the fixed points. The local contributions are in a form of a quotients with the local Chern classes in the numarators. On the other hand, as in the "residue theorem" for meromorphic functions, the sum of local Chern classes is equal to zero. Especially for Grassmanians we obtain interesting calculations with nontrivial formulas involving rational functions. We will discuss the issue of positivity: the local Chern class may be presented in a various ways, depending on some choice of a certain graphs. For some choices we find that the coefficients of the presentation are nonnegative. Also the coefficients in an appropriate Schur basis are nonnegative in many examples.
• Jaroslaw Wlodarczyk: Weights on the cohomology and invariants of singularities
  We study the weight filtration on the cohomology of a proper complex algebraic variety and obtain natural upper bounds on its size, when it is the exceptional divisor of a singularity. The invariants of singularities introduced here gives rather strong information about the topology of rational and related singularities. (based on joint paper with D.Arapura and P.Bakhtary)
• Jie Wu: Obtaining Higher Homotopy Groups from the Fundamental Groups
  The braid groups and link groups are the fundamental groups of configurations and link complements, respectively. In this talk, we give an illustration that the general higher homotopy groups of spheres can be obtained as the quotient groups of the intersections of certain canonical subgroups of these groups. The talk will give a philosophical view that the higher homotopy groups may be discovered by systematically studying the subgroups of the fundamental groups of canonical objects.
• Min Yan: Homotopy Classification of Multiaxial Actions
  A multiaxial action by U(n) is locally modeled on kC^n + C^p, where U(n) acts canonically on C^n and trivially on C^p. There are similar multiaxial actions by O(n) and Sp(2n). Mike Davis gave diffeomorphic classification of multiaxial smooth manifolds in case k \le n. We describe how to homotopically classify multiaxial topological manifolds without assuming k \le n. Moreover, we compute the classification in case of the multiaxial sphere. Such classification is a vast generalization of the classical result on fake complex projective spaces. In fact, up until now, not much is known about the homotopy classifications of actions by positive dimensional Lie groups. This is a joint work with S. Cappell and S. Weinberger.
• Shoji Yokura: Fiberwise bordism groups
  We introduce a notion of fiberwise bordism for certain families of smooth manifolds. We explain motivations for considering this bordism, some connections with other known results, etc. This is a joint work with M. Banagl and J. Schuermann.
• Masahiko Yoshinaga: Minimal Stratifications for Line Arrangements
  The homotopy type of complements of complex hyperplane arrangements have a special property, so called minimality (Dimca-Papadima and Randell, around 2000). In this talk, we introduce the "dual" object to the minimal CW complex for two dimensional real line arrangements, which we call minimal stratification. It is a real semialgebraic stratification which induces a partition into contractible manifolds. We also see associated presentation of the fundamental group. This talk is based on arXiv:1105.1857.
• Sergey Yuzvinsky: Topological complexity of arrangement complements
  Topological complexity of (motion planning on) a topological space has been defined by M.Farber as a specialization of Schwartz`s genus. Its calculation for hyperplane arrangement complements is important for topological robotics and relates to interesting problems involving the Orlik-Solomon algebras. We will survey old results on the topic and discuss some new ones.

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